Manuals/calci/MATRIXINVERSE

MATRIXINVERSE (a)

• $a$ is any matrix.

Description

• This function shows the inverse value of the given matrix.
• In Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle MATRIXINVERSE (a)} , $a$ is any square matrix.
• Inverse of a square matrix is also called reciprocal of a matrix and it is denoted by $A^{-1}$.
• Consider the square matrix A has an inverse which should satisfies the following condition $|A|\neq 0$
• Also $AA^{-1}=I$(Identity matrix).
• Consider 2x2 matrix:A=[a b;c d].
• The inverse of matrix A is calculated by

$A^{-1}={\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{-1}$=${\frac {1}{detA}}{\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}$ =${\frac {1}{ad-bc}}{\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}$

• Consider 3x3 matrix A and its inverse is calculated by

$A^{-1}={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}^{-1}$=${\frac {1}{detA}}{\begin{bmatrix}A&B&C\\D&E&F\\G&H&I\end{bmatrix}}^{T}$= ${\frac {1}{detA}}{\begin{bmatrix}A&D&G\\B&E&H\\C&F&I\end{bmatrix}}$

where A=(ei-fh); B=-(di-fg);C=(dh-eg); D=-(bi-ch); E=(ai-cg); F=-(ah-bg); G=(bf-ce)  H=-(af-cd);I=(ae-bd)